256 Some Elementary Statistical InferencesThusP(^12 <Y 3 )=∫∞1 / 2g 3 (y 3 )dy 3=∫ 11 / 224(y 35 −y^73 )dy 3 =243
256.Finally, the joint pdf of any two order statistics, sayYi<Yj, is easily expressed
in terms ofF(x)andf(x). We havegij(yi,yj)=∫yia···∫y 2a∫yjyi···∫yjyj− 2∫byj···∫byn− 1n!f(y 1 )×···×f(yn)dyn···dyj+1dyj− 1 ···dyi+1dy 1 ···dyi− 1.Since, forγ>0,
∫yx[F(y)−F(w)]γ−^1 f(w)dw = −[F(y)−F(w)]γ
γ∣
∣
∣
∣yx=[F(y)−F(x)]γ
γ,it is found thatgij(yi,yj)=⎧
⎨
⎩n!
(i−1)!(j−i−1)!(n−j)![F(yi)]i− (^1) [F(yj)−F(yi)]j−i− 1
×[1−F(yj)]n−jf(yi)f(yj) a<yi<yj<b
0elsewhere.
(4.4.3)
Remark 4.4.1(Heuristic Derivation). There is an easy method of remembering
the pdf of a vector of order statistics such as the one given in formula (4.4.3). The
probabilityP(yi<Yi<yi+Δi,yj<Yj<yj+Δj), where Δiand Δjare small,
can be approximated by the following multinomial probability. Innindependent
trials,i−1 outcomes must be less thanyi[an event that has probabilityp 1 =F(yi)
on each trial];j−i−1 outcomes must be betweenyi+Δiandyj[an event with
approximate probabilityp 2 =F(yj)−F(yi) on each trial];n−joutcomes must be
greater thanyj+Δj[an event with approximate probabilityp 3 =1−F(yj)oneach
trial]; one outcome must be betweenyiandyi+Δi[an event with approximate
probabilityp 4 =f(yi)Δion each trial]; and, finally, one outcome must be between
yjandyj+Δj[an event with approximate probabilityp 5 =f(yj)Δjon each trial].
This multinomial probability is
n!
(i−1)!(j−i−1)!(n−j)! 1! 1!
p 1 i−^1 p 2 j−i−^1 pn 3 −jp 4 p 5 ,
which isgi,j(yi,yj)ΔiΔj,wheregi,j(yi,yj) is given in expression (4.4.3).